Skip to main content
SpringerLink
Log in

Surface Singularities Appeared in the Hyperbolic Schwarz Map for the Hypergeometric Equation

  • Conference paper
Arrangements, Local Systems and Singularities

Part of the book series: Progress in Mathematics ((PM,volume 283))

  • 1015 Accesses

Abstract

Surface singularities, swallowtail and cuspidal edge, appear in the hyperbolic Schwarz map for the hypergeometric differential equation. Such singularities are studied in detail. After an overview of classical staffs, the hypergeometric equation and the Schwarz map, the hyperbolic Schwarz map is introduced. We study the singularities of this map, whose target is the hyperbolic 3-space, and visualize its image when the monodromy group is a finite group or a typical Fuchsian group. Several confluences of swallowtails are also observed.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, Singularities of differentiable maps, Vol. 1, Monographs in Math. 82, Birkhäuser, Boston, 1985.

    MATH  Google Scholar 

  2. K. Iwasaki, H. Kimura, S. Shimomura and M. Yoshida, From Gauss to Painlevé — A modern theory of special functions, Vieweg Verlag, Wiesbaden, 1991.

    MATH  Google Scholar 

  3. F. Klein, Vorlesungen über das Ikosaeder und die auflösung der Gleichungen vom fünften Grade, Teubner, Leipzig, 1884.

    Google Scholar 

  4. M. Kokubu, M. Umehara, and K. Yamada, “Flat fronts in hyperbolic 3-space”, Pacific J. Math. 216(2004), 149–175.

    Article  MATH  MathSciNet  Google Scholar 

  5. M. Kokubu, W. Rossman, K. Saji, M. Umehara, and K. Yamada, “Singularities of flat fronts in hyperbolic space”, Pacific J. Math. 221(2005), 303–351.

    Article  MATH  MathSciNet  Google Scholar 

  6. R. Langevin, G. Levitt and H. Rosenberg, “Classes d’homotopie de surfaces avec rebroussements et queues d’aronde dans R 3”, Can. J. Math. 47(1995), 544–572.

    MATH  MathSciNet  Google Scholar 

  7. M. Noro, T. Sasaki, K. Yamada and M. Yoshida, “Confluence of swallowtail singularites of the hyperbolic Schwarz map defined by the hypergeometric differential equation”, Experimental Math. 17(2008), 191–204.

    MATH  MathSciNet  Google Scholar 

  8. T. Sasaki, K. Yamada and M. Yoshida, “The hyperbolic Schwarz map for the hypergeometric differential equation”, Experimental Math. 17(2008), 269–282.

    MATH  MathSciNet  Google Scholar 

  9. T. Sasaki, K. Yamada and M. Yoshida, “Derived Schwarz map of the hypergeometric differential equation and a parallel family of flat fronts”, Internat. J. Math. 19(2008), 847–863.

    Article  MATH  MathSciNet  Google Scholar 

  10. M. Yoshida, Hypergeometric Functions, My Love, Vieweg Verlag, 1997.

    Google Scholar 

  11. M. Yoshida, “From the power function to the hypergeometric function”, Progress in Math., 260(2007), 407–429.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Kobe University, Kobe, 657-8501, Japan

    Takeshi Sasaki

  2. Faculty of Mathematics, Kyushu University, Fukuoka, 810-8560, Japan

    Masaaki Yoshida

Authors
  1. Takeshi Sasaki
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Masaaki Yoshida
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute of Mathematics, Case 7012, 2 place Jussieu, 75251, Paris Cedex 05, France

    Fouad El Zein

  2. Center for Advanced Mathematical Sciences (CAMS), American University of Beirut, P.O. Box 11-236, Beirut, Lebanon

    Fouad El Zein

  3. Department of Mathematics, Northeastern University, 360 Huntington Avenue, Boston, MA, 02115, USA

    Alexandru I. Suciu

  4. Mathematics Department, Galatasaray University, Ciragan Cad. No 36 Besiktas, 34357, Istanbul, Turkey

    Meral Tosun  & A. Muhammed Uludağ  & 

  5. Department of Mathematics, University of Oregon, Eugene, OR, 97405, USA

    Sergey Yuzvinsky

Additional information

Dedicated to Professor Fritz Hirzebruch on his eightieth birthday

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Birkhäuser Verlag Basel/Switzerland

About this paper

Cite this paper

Sasaki, T., Yoshida, M. (2009). Surface Singularities Appeared in the Hyperbolic Schwarz Map for the Hypergeometric Equation. In: El Zein, F., Suciu, A.I., Tosun, M., Uludağ, A.M., Yuzvinsky, S. (eds) Arrangements, Local Systems and Singularities. Progress in Mathematics, vol 283. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0209-9_10

Download citation

Publish with us

Policies and ethics

Search

Navigation